Axiom of countable choice

Each set in the countable sequence of sets (S_i) = S₁, S₂, S₃, ... contains a nonzero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (x_i) = x₁, x₂, x₃, ...

The axiom of countable choice or axiom of denumerable choice, denoted AC_ω, is an axiom of set theory that states that any countable collection of non-empty sets must have a choice function. I.e., given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, then there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.

The axiom of countable choice (AC_ω) is strictly weaker than the axiom of dependent choice (DC), (Jech 1973) which in turn is weaker than the axiom of choice (AC). Paul Cohen showed that AC_ω, is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice(Potter 2004). AC_ω holds in the Solovay model.

ZF + AC_ω suffices to prove that the union of countably many countable sets is countable. It also suffices to prove that every infinite set is Dedekind-infinite (equivalently: has a countably infinite subset).

AC_ω is particularly useful for the development of analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. For instance, in order to prove that every accumulation point x of a set S⊆R is the limit of some sequence of elements of S\{x}, one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to AC_ω. For other statements equivalent to AC_ω, see Herrlich (1997) and Howard & Rubin (1998).

A common misconception is that countable choice has an inductive nature and is therefore provable as a theorem (in ZF, or similar, or even weaker systems) by induction. However, this is not the case; this misconception is the result of confusing countable choice with finite choice for a finite set of size n (for arbitrary n), and it is this latter result (which is an elementary theorem in combinatorics) that is provable by induction. However, some countably infinite sets of nonempty sets can be proven to have a choice function in ZF without any form of the axiom of choice. These include V_ω− {Ø} and the set of proper and bounded open intervals of real numbers with rational endpoints.

Use

As an example of an application of AC_ω, here is a proof (from ZF+AC_ω) that every infinite set is Dedekind-infinite:

Let X be infinite. For each natural number n, let A_n be the set of all 2ⁿ-element subsets of X. Since X is infinite, each A_n is nonempty. A first application of AC_ω yields a sequence (B_n : n=0,1,2,3,...) where each B_n is a subset of X with 2ⁿ elements.

The sets B_n are not necessarily disjoint, but we can define

C₀ = B₀

C_n= the difference of B_n and the union of all C_j, j<n.

Clearly each set C_n has at least 1 and at most 2ⁿ elements, and the sets C_n are pairwise disjoint. A second application of AC_ω yields a sequence (c_n: n=0,1,2,...) with c_n∈C_n.

So all the c_n are distinct, and X contains a countable set. The function that maps each c_n to c_n+1 (and leaves all other elements of X fixed) is a 1-1 map from X into X which is not onto, proving that X is Dedekind-infinite.

References

This article incorporates material from axiom of countable choice on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.